Algebra Examples

$\log (2 x + 1) + \log (x - 4) = \log (2 x^{2} - 4)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log ((2 x + 1) (x - 4)) = \log (2 x^{2} - 4)$

Step 1.2

Expand $(2 x + 1) (x - 4)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$\log (2 x (x - 4) + 1 (x - 4)) = \log (2 x^{2} - 4)$

Step 1.2.2

Apply the distributive property.

$\log (2 x \cdot x + 2 x \cdot - 4 + 1 (x - 4)) = \log (2 x^{2} - 4)$

Step 1.2.3

Apply the distributive property.

$\log (2 x \cdot x + 2 x \cdot - 4 + 1 x + 1 \cdot - 4) = \log (2 x^{2} - 4)$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.1.1

Move $x$ .

$\log (2 (x \cdot x) + 2 x \cdot - 4 + 1 x + 1 \cdot - 4) = \log (2 x^{2} - 4)$

Step 1.3.1.1.2

Multiply $x$ by $x$ .

$\log (2 x^{2} + 2 x \cdot - 4 + 1 x + 1 \cdot - 4) = \log (2 x^{2} - 4)$

Step 1.3.1.2

Multiply $- 4$ by $2$ .

$\log (2 x^{2} - 8 x + 1 x + 1 \cdot - 4) = \log (2 x^{2} - 4)$

Step 1.3.1.3

Multiply $x$ by $1$ .

$\log (2 x^{2} - 8 x + x + 1 \cdot - 4) = \log (2 x^{2} - 4)$

Step 1.3.1.4

Multiply $- 4$ by $1$ .

$\log (2 x^{2} - 8 x + x - 4) = \log (2 x^{2} - 4)$

Step 1.3.2

Add $- 8 x$ and $x$ .

$\log (2 x^{2} - 7 x - 4) = \log (2 x^{2} - 4)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$2 x^{2} - 7 x - 4 = 2 x^{2} - 4$

Step 3

Solve for $x$ .

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Step 3.1

Move all terms containing $x$ to the left side of the equation.

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Step 3.1.1

Subtract $2 x^{2}$ from both sides of the equation.

$2 x^{2} - 7 x - 4 - 2 x^{2} = - 4$

Step 3.1.2

Combine the opposite terms in $2 x^{2} - 7 x - 4 - 2 x^{2}$ .

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Step 3.1.2.1

Subtract $2 x^{2}$ from $2 x^{2}$ .

$- 7 x - 4 + 0 = - 4$

Step 3.1.2.2

Add $- 7 x - 4$ and $0$ .

$- 7 x - 4 = - 4$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Add $4$ to both sides of the equation.

$- 7 x = - 4 + 4$

Step 3.2.2

Add $- 4$ and $4$ .

$- 7 x = 0$

Step 3.3

Divide each term in $- 7 x = 0$ by $- 7$ and simplify.

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Step 3.3.1

Divide each term in $- 7 x = 0$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{0}{- 7}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 7$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{0}{- 7}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 7}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $0$ by $- 7$ .

$x = 0$

Step 4

Exclude the solutions that do not make $\log (2 x + 1) + \log (x - 4) = \log (2 x^{2} - 4)$ true.

$No solution$